Field of the Invention
This invention relates generally to the field of X-ray diffraction and, more specifically, to the compensation for scattering angle distortions in two-dimensional X-ray detectors.
Description of the Related Art
In the field of x-ray diffraction, radiation with a wavelength, λ, in the subnanometer range is directed to a crystalline material with a given interatomic spacing, d. When the angle of incidence, θ, relative to the crystalline structure satisfies the Bragg equation, λ=2d sin θ, an interferometrically reinforced signal (the diffracted signal), may be observed leaving the material, with an angle of emission being equal to an angle of incidence, both angles being measured with respect to a direction normal to the interatomic spacing of interest.
Diffracted X-rays from a single crystal sample follow discrete directions each corresponding to a family of diffraction planes, as shown schematically in FIG. 1A. The diffraction pattern from a polycrystalline (powder) sample forms a series diffraction cones, as shown in FIG. 1B, if large numbers of crystals oriented randomly in the space are covered by the incident x-ray beam. Each diffraction cone corresponds to the diffraction from the same family of crystalline planes in all the participating grains. Polycrystalline materials can be single-phase or multi-phase in bulk solid, thin film or fluid. For example, FIG. 2 shows the diffraction pattern of corundum powder collected by a two-dimensional (2D) X-ray detector.
FIG. 3 is a schematic illustration showing the geometry of an X-ray diffraction system in the laboratory coordinates system XLYLZL. The origin of the coordinate system is the instrument center, or goniometer center. The source X-ray beam propagates along the XL axis, which is also the rotation axis of the diffraction cones. The apex angles of the diffraction cones are determined by the 2θ values given by the Bragg equation. In particular, the apex angles are twice the 2θ values for forward reflection (2θ≦90°) and twice the values of 180°-2θ for backward reflection (2θ>90°). The XL-YL plane is the diffractometer plane. The γ angle defines the direction of a diffracted beam relative to the diffraction cone. It is measured within a plane parallel to the YL-ZL plane from the point at which the cone intersects the −z portion of the y=0 axis to the point at which the diffracted beam intersects the plane. Thus, a point in the −YL portion of the diffractometer plane corresponds to γ=90°, while a point in the +YL portion of the diffractometer plane corresponds to γ=270°. Thus, the γ and 2θ angles form a kind of spherical coordinate system which covers all the directions from the origin, where the sample is located.
An ideal detector to measure a diffraction pattern in three-dimensional (3D) space is a detector with spherical detecting surface, as shown in FIG. 4. The sample is in the center of the sphere, and all of the pixels of the detector are equally distanced from the sample. This configuration, however, is very impractical and, in practice, the detection surface will be flat, cylindrical or with another curved shape. Therefore, the pixel-to-sample distance varies within a detector. Correspondingly, angular coverage of a pixel in the same detector is different depending on the location of the pixel within the detector.